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LETTER TO THE EDITOR
The following is the abstract of the article discussed in the subsequent letter:
The presence of negatively charged, impermeant proteins in the plasma space alters the distribution of diffusible ions in the plasma and interstitial fluid (ISF) compartments to preserve electroneutrality. We have derived a new mathematical model to define the quantitative interrelationship between the Gibbs-Donnan equilibrium, the osmolality of body fluid compartments, and the plasma water Na+ concentration ([Na+]pw) and validated the model using empirical data from the literature. The new model can account for the alterations in all ionic concentrations (Na+ and non-Na+ ions) between the plasma and ISF due to Gibbs-Donnan equilibrium. In addition to the effect of Gibbs-Donnan equilibrium on Na+ distribution between plasma and ISF, our model predicts that the altered distribution of osmotically active non-Na+ ions will also have a modulating effect on the [Na+]pw by affecting the distribution of H2O between the plasma and ISF. The new physiological insights provided by this model can for the first time provide a basis for understanding quantitatively how changes in the plasma protein concentration modulate the [Na+]pw. Moreover, this model defines all known physiological factors that may modulate the [Na+]pw and is especially helpful in conceptually understanding the pathophysiological basis of the dysnatremias.
Master equation for dysnatremia or intractable abracadabra
To the Editor: Nguyen and Kurtz in their recent paper (5) state that sodium concentration ([Na]) can be computed from a formula praised in an invited editorial as the master dysnatremia equation.
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In spite of these minor quibbles, the authors fortunately validate the model using empirical data in the literature. This amounts to tabulated osmolality data from Guyton, which are then used on a (hypothetical?) 70-kg male with plasma volume of 3 liters, interstitial volume of 14 liters, and intracellular volume of 25 liters. These volumes may be a bit atypical [at least Edelman and Leibman (2) earlier summarized measurements yielding 120 ml/kg of interstitial fluid corresponding to 8.4 liters]. Fortunately this disagreement of a few liters has no consequence because using the formula provided by Nguyen and Kurtz (Eq. no. 11) and keeping all the osmolalities and total body water and letting the interstitial fluid vary between 1 and 40 liters recovers the initial [Na] of 142 in every case.
To help us appreciate the insights gained with this new master equation, the authors examine some important situations. Hence the effect of urea is scrutinized by hypothetically letting the data from Guyton being augmented by 10 mosM of urea. This does not move water, we know, and hence everywhere in the equation the osmotic effect is proportionally increased while volumes are unchanged, and obviously it all balances in the end and does not move water as stated initially.
The real crux of the paper is probably the effect of the Gibbs-Donnan equilibrium, and to prove the important effect on [Na] of varying the Gibbs-Donnan equilibrium the authors give two references. In the paper by Dandona et al. (1), improvement in hyponatremia was seen ensuing albumin infusion. According to Nguyen and Kurtz (5) the urine excretion of free water could not account for this because urine osmolality was still increased, and so the present authors inferred that the effect was caused by the Gibbs-Donnan equilibrium. Salt-free infusions were not used in the paper (1), but even more to the point, it is amazing that the present authors seem to believe that urine osmolality governs the renal effect on plasma sodium given the otherwise prevalent thinking, since Edelman et al. (3) and Rose (8), that it is the electrolyte-free water clearance. The other retrospective paper (1) describes a heterogeneous sample of ill patients with hyponatremia and hypoalbuminemia. It is utterly impossible to know how these hypos are related apart from a statistically significant correlation of 0.43 indicating that 
16% of the variability in [Na] is "explained" by albumin. The fact that this paper is cited to prove the concept of the Gibbs-Donnan equilibration to influence the regulation of [Na] may indicate that the concept is altogether weakly founded.
Finally, Nguyen and Kurtz (5) also give us a formula helping us compute the expected changes in [Na] when maltose is infused, which is 0.59 mmol/l for each millimolar increase in maltose, as stated in the paper by Palevsky et al. (6). Strangely, however, the present authors fail to inform us that the reason for infusing maltose in the first place was the need to administer 80 g of immunoglobulin IV because of thrombocytopenia. One wonders why this huge amount of protein goes unmentioned, especially in this connection. At any rate it should influence the correction factor for maltose, at least if Gibbs-Donnan is not severely allergic to maltose, a concept yet to be developed.
I am sure these and other problems will be covered in the next set of equations from the present authors.
REFERENCES
To the Editor: In his Letter to the Editor, Ring criticizes our new formula (Eq. 12) defining the quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality of body fluid compartments, and plasma water sodium concentration (11) as being too difficult to use. Ring argues that the measurement of exchangeable Na+ and K+ by isotope dilution is a tedious process and that osmotically inactive Na+ and K+ are difficult to quantify. First, the significance of Eq. 12 should not be based on its simplicity of usage, but rather on the extent to which it accurately models human physiology. In this regard, the parameters in Eq. 12 accurately define the quantitative interrelationship of all the known physiological factors modulating the plasma water Na+ concentration ([Na+]pw). Second, from the standpoint of the [Na+]pw, only the osmotically active Na+ and K+ need to be considered. Because the total exchangeable Na+ (Nae) and total exchangeable K+ (Ke) include osmotically active as well as osmotically inactive Na+ and K+, respectively, the terms (Nae + Ke)/TBW – (Naosm inactive + Kosm inactive)/TBW in Eq. 12 represent the osmotically active Na+ and K+ ions. Currently, the determination of the [Na+] and [K+] by our conventional laboratory techniques measures only the osmotically active Na+ and K+ ions rather than the exchangeable bound, osmotically inactive Na+ and K+ ions (e.g., exchangeable bound, osmotically inactive Na+ in bone); i.e., our conventional laboratory techniques do not involve isotope dilution and therefore do not measure total exchangeable Na+ and total exchangeable K+. Therefore, it is irrelevant that Nae, Ke, Naosm inactive, and Kosm inactive cannot be easily measured because our current laboratory techniques measure osmotically active Na+ and K+ directly [as represented by the terms (Nae + Ke)/TBW – (Naosm inactive + Kosm inactive)/TBW in Eq. 12].
Ring also questions recent studies (6, 16, 17) suggesting that there is osmotically inactive Na+ storage during clinical states of Na+ retention. We agree with Ring that, on the basis of these studies (6, 16, 17), it cannot be concluded that the osmotically inactive Na+ storage pool is dynamically regulated during Na+ retention because these studies do not account for both Na+ and K+ balance. Indeed, by considering the mass balance of Na+, K+, and H2O, Seeliger et al. (15) demonstrated that Na+ accumulation occurs in an osmotically active form during Na+ retention. However, although the findings of the latter study suggest that the osmotically inactive Na+ storage pool is not dynamically regulated during Na+ retention, it does not provide any evidence for or against the existence of a fixed pool of osmotically inactive Na+ storage.
We disagree with Ring's assertion that "it may be a little daring play with words to state that exchangeable sodium equates with osmotically active and osmotically inactive sodium outside a concrete experiment explaining exactly how long (isotope) dilution is done over." First, the duration of radioisotope dilution only determines the completeness of radioisotope equilibration and therefore defines the fraction of the total body Na+ pool that is exchangeable and nonexchangeable. The duration of radioisotope dilution does not provide any information as to whether the Nae pool is osmotically active or inactive. Rather, the osmotic activity of Nae depends on its ability to move randomly in solution. Therefore, any factor that reduces the random movement of Na+ will reduce its osmotic activity. For example, the osmotic activity of Nae in plasma is slightly less than one because the electrostatic interactions between ions reduce the random movement of Na+ (8). The osmotic coefficient Ø in Eq. 12 therefore accounts for the activity of Na+ salts as independent osmotically active particles under physiological conditions (8). Similarly, not all Nae is osmotically active because a portion of Nae is bound in bone (3, 4). Because the random movement of exchangeable Na+ bound to bone is significantly reduced, exchangeable Na+ bound to bone is therefore rendered osmotically inactive. The component of Nae that is osmotically inactive in bone is determined by calculating the ratio of the specific activities of Na+ in bone and plasma (3).
Ring also criticizes that our estimate of the interstitial fluid (ISF) volume (14) is "atypical" simply because it is different from that measured by Edelman and Leibman (2). However, this criticism is irrelevant because our mathematical model will accurately predict the correct [Na+]pw regardless of what estimate of plasma, ISF, and intracellular fluid volumes one use to validate the model. In fact, Ring admittedly acknowledges that our formula accurately predicts the correct [Na+]pw when the ISF volume is varied from 1 to 40 liters.
Regarding the study of Dandona et al. (1), Ring argues that salt-free albumin infusions were not used in the study and therefore may contribute to the correction of the hyponatremia. Although the [Na+] in 25% human albumin is typically 145 ± 15 mmol/l (7), it is overly simplistic to suggest that the Na+ content in the small volume of albumin infusions would lead to a significant correction of the hyponatremia. In this regard, it has been shown quantitatively that isotonic saline ([Na+] = 154 mmol/l) has a minor direct effect on the correction of hyponatremia (9). Rather, the reason that isotonic saline has a role in the correction of hyponatremia is due to its suppressive effect on antidiuretic hormone (ADH) secretion. In other words, correction of hyponatremia occurs as a result of the renal excretion of electrolyte-free water once the hypovolemic stimulus to ADH secretion is removed with volume replacement.
We disagree with Ring's inference that we believe that the rate of urinary free water excretion is better assessed by urinary osmolality rather than urinary electrolyte-free water clearance. We are fully aware that urea is a component of the measured urinary osmolality and that urea does not alter the [Na+]pw by modulating the distribution of water between the body fluid compartments. Indeed, we recently derived a new formula termed modified electrolyte-free water clearance to accurately determine the urinary electrolyte-free water clearance (10). Given that there are insufficient data in the Dandona et al.'s study to calculate the urinary electrolyte-free water clearance, one has to rely on the available data on urinary osmolality to interpret the findings of this study. Although urinary osmolality has its limitation, it still can provide useful information if interpreted correctly. In the study, the authors attributed the improvement in the hyponatremia with albumin infusions to the nonosmotic suppression of ADH secretion. However, if ADH secretion were to be shut off because of removal of the hypovolemic stimulus, then the urinary osmolality would be expected to be less than 100 mosmol/kgH2O in these hyponatremic patients (14). Therefore, the marked improvement in the hyponatremia in these patients cannot be due to a significant increase in the urinary free water excretion since the urinary osmolality remained inappropriately elevated postalbumin infusion in these hyponatremic patients (urine osmolality ranged from 350 to 430 mosmol/kgH2O postalbumin infusion). Ring seems to have missed this important point.
Regarding the study of Ferreira da Cunha et al. (5), we referenced this paper only to demonstrate that there is an association between hypoalbuminemia and hyponatremia. As clearly stated in our paper, we made no other conclusions from the findings of this study. However, we disagree with Ring's assertion that the concept of Gibbs-Donnan equilibrium in influencing "the regulation of the [Na+]" may be "altogether weakly founded." Indeed, it is well known that the [Na+]pw is greater than the interstitial fluid sodium concentration owing to the Gibbs-Donnan equilibrium (13). In fact, the Gibbs-Donnan ratio for the distribution of Na+ ions between the plasma and interstitial fluid is 100:95 (13).
We disagree with Ring that intravenous immunoglobulin (IVIG) contributes significantly to the Gibbs-Donnan equilibrium. For a charged particle to exert the Gibbs-Donnan effect, it has to be nonpermeant and restricted to a given compartment (12, 13). Because IVIG is evenly distributed between the intravascular and extravascular spaces (7), it is unlikely that IVIG contributes to the Gibbs-Donnan equilibrium. Moreover, Ring is incorrect in implicating that Gibbs-Donnan equilibrium modulates the [Na+]pw by altering the distribution of maltose. It is well known that the distribution of noncharged particles (e.g., maltose) is unaltered by the Gibbs-Donnan effect. Ring seems to be unaware that the physiological basis underlying Gibbs-Donnan equilibrium is the establishment of electrochemical equilibrium, and therefore only the distribution of charged particles is affected.
Finally, in response to Ring's suggestion that Eq. 12 is simply "intractable abracadabra," the fundamental point to be emphasized is that physiological factors other than Na+, K+, and H2O also have a modulating effect on the [Na+]pw (11). Importantly, the complex interplay of these various factors modulating the [Na+]pw resulting from Gibbs-Donnan and osmotic equilibrium demonstrates how Eq. 12 can be an indispensable tool in the quantitative understanding and analysis of the pathophysiology of the dysnatremias.
REFERENCES
This article has been cited by other articles:
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  K. L. Dorrington Master dysnatremia equation for Gibbs-Donnan equilibrium and plasma sodium concentration: proof or spoof? J Appl Physiol, February 1, 2008; 104(2): 569 - 569. [Full Text] [PDF]
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